Shifted Bernstein Polynomial-Based Dynamic Analysis for Variable Fractional Order Nonlinear Viscoelastic Bar

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

According to the author, this study presents a shifted Bernstein polynomial-based method for numerically solving the variable-fractional-order control equation governing a viscoelastic bar. Initially, by employing a variable-order fractional constitutive relation alongside the equation of motion, the control equation for the viscoelastic bar is derived. Shifted Bernstein polynomials serve as basis functions for approximating the bar’s displacement function, and the variable fractional derivative operator matrix is developed. Subsequently, the displacement control equation of the viscoelastic bar is transformed into a matrix product. Discretization converts the operator matrix into a solvable algebraic system, enabling the direct numerical solution of the displacement of the variable fractional viscoelastic bar control equation in the time domain. Additionally, a convergence analysis is performed. Finally, the algorithm's precision and efficacy are confirmed through computations.

 

The paper is well-written, and the results appear to be original. However, the paper requires major revisions before it can be considered for publication in this esteemed journal.

The main equation presented in (9) is derived from the article [37] and is solved using a numerical method. The authors must clarify their scientific contribution to this work.

 

Definition 1 must be properly cited with reference to the original article. The current citation appears to be incorrect.

Introduction Section: The authors discuss various methods for solving complex problems, such as the multiscale method [19], Galerkin method [20], finite element method [21], homotopy perturbation method [22], variational iteration method [23], and Adomian decomposition method [24]. However, they should also discuss meshless methods, which are powerful tools for such problems. For example, they may consider: Numerical solution of two-term time-fractional PDE models arising in mathematical physics using a local meshless method.

The authors mention methods involving Chebyshev polynomials [26], Legendre polynomials [27], wavelet functions [28], and Bernstein polynomial-based methods [29–32]. A recently published paper using Lucas polynomials could be relevant: An Efficient Numerical Solution of a Multi-Dimensional Two-Term Fractional Order PDE via a Hybrid Methodology: The Caputo–Lucas–Fibonacci Approach with Strang Splitting, Fractal and Fractional.

The numerical results presented in Figure 2 need verification.

The numerical results shown in the tables are computed for a short final time (e.g., T=0.8). The authors should analyze the method’s behavior for longer final times, such as T=5 or T=10.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Review on the paper “Shifted Bernstein Polynomial-Based Dynamic Analysis for Variable Fractional Order Nonlinear Viscoelastic Bar” by Zhongze Li, Lixing Ma, Yiming Chen, Jingguo Qu, Yuhuan Cui and Lei Wang.

 

The paper is connected with numerical study of the behavior of solutions to variable-fractional-order control equation describing a viscoelastic bar. The authors derived a new control equation for the viscoelastic bar by means of a variable-order fractional constitutive relation alongside the equation of motion. Using the FEM method, the authors constructed new basis-functions to approximate the bar’s displacement function and the variable fractional derivative operator matrix. Thus, they transformed the displacement control equation of the viscoelastic bar into a matrix product and get a solvable algebraic system, which in turn allows the numerical solution of the displacement of the variable fractional viscoelastic bar control equation to be solved. As a result, the authors presented a convergence analysis.

The introduction to the article is clear, detailed enough and the analysis of the known literature is excellent. It should be also noted that the figures illustrate the text very well. The conclusion correctly reflects the results of the paper.

I think the results are interesting and new. The article, after minor revision, can be published in a journal.

 

Comments.

For convenience of readers, it is better to use $\int\limits$ instead of $\int$ for integrals.

It would be interesting to understand the choice of definition of fractional derivative. The authors chose Caputo's definition, although they could have used Riemann-Liouville's definition or the definition using Fourier transform. A remark on this should be made.

Instead of “4.1. Approximation of function” it is better to write “4.1. Approximation of functions”.

It is advisable to remove the reference inside Theorem 1. It is better to place it before the formulation of the theorem.

Formula (40), lines 190-192. It is necessary to explain the appearance of such precise constants in the equation. Either they are from applications, or simply artificially assigned constants.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have incorporated all the suggestions, and the article has significantly improved. Therefore, I recommend it for publication in this esteemed journal.

Author Response

We sincerely appreciate the reviewers' positive feedback and recommendation for publication. Thank you for your support throughout this process.